3.100.52 \(\int \frac {11 x+10 x^2+(10+10 x) \log (-3 x)}{x} \, dx\)

Optimal. Leaf size=13 \[ -4+x+5 (x+\log (-3 x))^2 \]

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Rubi [A]  time = 0.04, antiderivative size = 22, normalized size of antiderivative = 1.69, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {14, 2346, 2301, 2295} \begin {gather*} 5 x^2+x+5 \log ^2(-3 x)+10 x \log (-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(11*x + 10*x^2 + (10 + 10*x)*Log[-3*x])/x,x]

[Out]

x + 5*x^2 + 10*x*Log[-3*x] + 5*Log[-3*x]^2

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (11+10 x+\frac {10 (1+x) \log (-3 x)}{x}\right ) \, dx\\ &=11 x+5 x^2+10 \int \frac {(1+x) \log (-3 x)}{x} \, dx\\ &=11 x+5 x^2+10 \int \log (-3 x) \, dx+10 \int \frac {\log (-3 x)}{x} \, dx\\ &=x+5 x^2+10 x \log (-3 x)+5 \log ^2(-3 x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 22, normalized size = 1.69 \begin {gather*} x+5 x^2+10 x \log (-3 x)+5 \log ^2(-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(11*x + 10*x^2 + (10 + 10*x)*Log[-3*x])/x,x]

[Out]

x + 5*x^2 + 10*x*Log[-3*x] + 5*Log[-3*x]^2

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fricas [A]  time = 0.89, size = 22, normalized size = 1.69 \begin {gather*} 5 \, x^{2} + 10 \, x \log \left (-3 \, x\right ) + 5 \, \log \left (-3 \, x\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+10)*log(-3*x)+10*x^2+11*x)/x,x, algorithm="fricas")

[Out]

5*x^2 + 10*x*log(-3*x) + 5*log(-3*x)^2 + x

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giac [A]  time = 0.15, size = 22, normalized size = 1.69 \begin {gather*} 5 \, x^{2} + 10 \, x \log \left (-3 \, x\right ) + 5 \, \log \left (-3 \, x\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+10)*log(-3*x)+10*x^2+11*x)/x,x, algorithm="giac")

[Out]

5*x^2 + 10*x*log(-3*x) + 5*log(-3*x)^2 + x

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maple [A]  time = 0.02, size = 23, normalized size = 1.77

method	result	size
derivativedivides	\(10 \ln \left (-3 x \right ) x +x +5 x^{2}+5 \ln \left (-3 x \right )^{2}\)	\(23\)
default	\(10 \ln \left (-3 x \right ) x +x +5 x^{2}+5 \ln \left (-3 x \right )^{2}\)	\(23\)
norman	\(10 \ln \left (-3 x \right ) x +x +5 x^{2}+5 \ln \left (-3 x \right )^{2}\)	\(23\)
risch	\(10 \ln \left (-3 x \right ) x +x +5 x^{2}+5 \ln \left (-3 x \right )^{2}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((10*x+10)*ln(-3*x)+10*x^2+11*x)/x,x,method=_RETURNVERBOSE)

[Out]

10*ln(-3*x)*x+x+5*x^2+5*ln(-3*x)^2

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maxima [A]  time = 0.35, size = 22, normalized size = 1.69 \begin {gather*} 5 \, x^{2} + 10 \, x \log \left (-3 \, x\right ) + 5 \, \log \left (-3 \, x\right )^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+10)*log(-3*x)+10*x^2+11*x)/x,x, algorithm="maxima")

[Out]

5*x^2 + 10*x*log(-3*x) + 5*log(-3*x)^2 + x

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mupad [B]  time = 7.12, size = 22, normalized size = 1.69 \begin {gather*} 5\,x^2+10\,x\,\ln \left (-3\,x\right )+x+5\,{\ln \left (-3\,x\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((11*x + 10*x^2 + log(-3*x)*(10*x + 10))/x,x)

[Out]

x + 10*x*log(-3*x) + 5*log(-3*x)^2 + 5*x^2

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sympy [A]  time = 0.10, size = 26, normalized size = 2.00 \begin {gather*} 5 x^{2} + 10 x \log {\left (- 3 x \right )} + x + 5 \log {\left (- 3 x \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((10*x+10)*ln(-3*x)+10*x**2+11*x)/x,x)

[Out]

5*x**2 + 10*x*log(-3*x) + x + 5*log(-3*x)**2

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